Debby

One failure in eight costumes

Eight positions, six disciplines, about fifteen hundred years, and no contact between most of them. They all arrive at the same structural fact: knowing how many roles a relationship has does not tell you how many individuals are in it. Three of them route around it, and all three route around it the same way.

How to read this

The centre is the finding, stated as a sentence: An inventory of kinds is not a count of individuals. A relation can have exactly three places and still be satisfied by one thing filling all three. Each spoke is a position that arrives there, placed clockwise from the top in date order, so going once round the wheel is going through the chronology. 5 of them collide with the centre — they establish the shape of a relationship and stop. 3 route around it, and all 3 do it the same way: not by counting anything, but by saying what a particular relation is.

Collides — establishes a shape, and stopsRoutes around — answers with what a relation isClockwise from the top is chronological.
  1. c. 480

    Neoplatonic metaphysics

    Proclus

    A derivation has exactly three stages — remaining, procession, return — and the return closes the circuit, so no fourth is needed.

    CollidesThree stages of one process. The case analysis never asks how many individuals run through them.

  2. early 1100s

    Byzantine arithmology

    Eustratius of Nicaea

    The Son and the Spirit must be perfectly co-ordinate: an isosceles triangle with the Father at the apex.

    CollidesThe symmetry the argument needs is a symmetry that makes the two base positions indiscernible — and indiscernible positions collapse.

  3. before 1173

    Medieval theology

    Richard of St Victor

    Charity cannot be self-directed; mutual love between two is not shared love; shared love needs a third.

    Routes aroundRoutes around it. The premise is an irreflexive relation, not a count of places. He is exposed on the ascent instead — why a pair must be better than one.

  4. c. 1268

    Scholastic theology

    Thomas Aquinas

    Four real relations in God and three persons: relations distinguish persons only where they stand opposed.

    Routes aroundRoutes around it, and denies it outright. The tradition's most rigorous exponent gets three from four and says why.

  5. c. 1903

    Formal logic

    Charles Sanders Peirce

    Three-place relations cannot be built from two-place ones, and anything larger can be built from three-place ones.

    CollidesHis own teridentity is an irreducible triad with exactly one occupant. The proof and the counterexample share an author.

  6. 1994

    Analytic theology

    Richard Swinburne

    A perfectly loving God must share love, and shared love requires a third person to be loved by two.

    Routes aroundRoutes around it by the same irreflexivity as Richard — then stops at three with a silence rather than a barrier.

  7. 2001

    Philosophy of mathematics

    Jukka Keränen

    Structures routinely contain positions that nothing inside the structure can tell apart.

    CollidesEvery structuralist account of identity then makes those positions the same thing — so the theory says 1 = −1. Structure collapses occupants; it does not multiply them.

  8. 2013–2018

    Metaphysics of relations

    Gilmore and Dixon

    A universal is three-placed exactly when there are three slots in it. Argument-places are real entities.

    CollidesThe theory's own axioms let one object occupy more than one slot at once. The field states the denial as a starting assumption.